IA2-SE

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Oct 30, 2023

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MAGNETIC FIELD STRENGTH WITH DISTANCE RATIONALE Exercise 12.2 was conducted to deduce how force varies with current within a magnetic field. According to: F = BI l ⊥ Equation 1: Force on a current in a magnetic field it was expected that force (F) was directly proportional to current (I). A plot of force against current was formulated from the data and a positive linear correlation between each variable was determined. It was observed that the magnetic field strength (B) was particularly sensitive to the placement of the magnets on the aluminium track, leading to the idea that B could have a distinct relationship with the separation (r) between the poles of two magnets. Consider the case of a magnetic dipole between two magnets, demonstrated below: Figure 1: Magnetic field between two magnets (Texas gateway, 2021) When the poles are brought closer the field lines join together and become denser, hence the field becomes stronger. Therefore, the opposite effect would occur if the poles were brought further apart (Texas Gateway, 2021). IA2-SE 1

It is known that the strength of the magnetic field from the face of a single dipole magnet, measured along its axis can be represented as: B axis = μ 0 4 π 2 μ d 3 Equation 2: Theoretical Field Strength Calculation (Vernier, 2020) The permeability constant ( μ 0 ) is equal to 4 π 10 − 7 T m / A (Vernier, 2020) and the magnetic moment ( μ ) for the magnet is equal to 8.09 A m 2 , assuming a grade of N42 (K&J magnets, 2021). By substituting the appropriate values into equation 2, a plot of B vs r can be formulated: 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.000 0.200 0.400 0.600 0.800 1.000 1.200 1.400 1.600 1.800 f(x) = 0 x^-3 R² = 1 Separation r (m) Magnetic Field Strength B (T) Figure 2: Theoretical B vs r Figure 2 indicates that B decreases as r is increased, in an inverse cube manner. However, this relationship is expected for the field from a single magnet and therefore does not reflect a situation with two magnets. Therefore, the relationship between B and r between two magnets could be reduced from an inverse power of 3 and expected to be within a range of B ∝ 1 r →B ∝ 1 r 3 considering that each field will interact with each other. IA2-SE 2

In attempt to experimentally investigate the relationship between B and r , exercise 12.2 will be redirected to measure the force, for different distances of separation between two neodymium magnets with a constant current flowing between them, using a constant length of wire and voltage. Since B cannot be measured directly with classroom apparatus, the experimental data was substituted into the following equation: B = F I .l ⊥ Equation 3: Magnetic field strength across a current-carrying wire A plot of B (T) against r (m) will be formulated from the processed data to determine a power relationship. This plot will be finally linearised to calculate a mathematical relationship for the experiment. RESEARCH QUESTION What is the relationship between the magnetic field strength (B) in-between the poles of two cylindrical neodymium magnets, of diameter 18mm and length 27mm and the distance of separating them, ranging from 9.82mm and 35.35mm, measured using a probe balance with a controlled length of 0.0088m, voltage of 2 V and current of 0.600 A? METHOD ORIGINAL METHOD Exercise 12.2 was conducted to test the dependency of force on the current flowing through a wire of constant length between the poles of two neodymium magnets, using an electronic balance and rheostat to control the current recorded by a digital multimeter. A graph of F vs I was established, and it was found that F ∝ I , which satisfied the theory relating to equation 1 (Nelson prac 8.4.3, 2018). IA2-SE 3

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MODIFICATIONS The experiment was redirected by measuring the major dependency of magnetic field strength on the distance separating the poles of the magnets, while controlling the current, length of wire and voltage. Force was the minor dependant variable in this experiment as it was required to calculate B . Aluminium plates were introduced to change the distance between the magnets to ensure that the wire was always at the centre of the field. A vernier caliper was also used to measure the separation (mm) instead of a ruler, in order to reduce the uncertainty in the distance values. Voltage was decreased from 6.00 V to 2.00 V to ensure the rheostat would not overheat. Four force readings were measured for 9 distances rather than 2, to reduce the percentage uncertainty in the average force. Finally, all measurements were taken using the same magnets over two days to ensure a controlled environment. DIAGRAM OF SETUP IA2-SE 4 Power pack Digital multimeter Rheostat Copper wire Retort Stand Neodymium magnets Aluminium plates Aluminium Tracks Electronic balance

MANAGEMENT OF RISKS Note: No environmental or ethical risks were present in this experiment Consideration Identification Management Neodymium magnets Fragments of material are fired off when magnets collide. Magnets colliding on skin or soft tissue will cause cuts and bruises. Ensure that one of the magnets is secure, so that the other can be used to test which face causes attraction. Wear eye protection. Rheostat Overheats when current is above 1 amp. Can cause burns to the fingers and hands. Ensure current is kept below 1 amp and voltmeter is turned on and off frequently. Power Pack Possibility of electric shock – level of risk depends on output voltage. Ensure leads are well insulated at each point therefore impossible to touch live metal. Table 1: Risk management RAW DATA Mass m (g) ± 0.005 Separation r (mm) ± 0.005 mm T1 T2 T3 T4 Average m (g) ± 9.82 0.59 0.58 0.59 0.51 0.57 0.04 13.03 0.28 0.30 0.29 0.41 0.32 0.07 15.92 0.34 0.28 0.35 0.35 0.33 0.04 19.21 0.18 0.18 0.17 0.25 0.20 0.04 22.22 0.19 0.18 0.23 0.20 0.20 0.03 25.57 0.12 0.12 0.12 0.16 0.13 0.02 28.89 0.11 0.12 0.10 0.15 0.12 0.03 31.74 0.08 0.09 0.08 0.11 0.09 0.02 35.35 0.07 0.08 0.07 0.10 0.08 0.02 Four force readings were measured for 9 different distances. The average and absolute uncertainty were then calculated. Table 2: Raw Data The average mass showed a gradual decrease as the separation was increased. Mass m (kg) ± 0.000005 kg Separation r (m) ± 0.0005 m T1 T2 T3 T4 Average m (kg) ± IA2-SE 5

9.82E-03 5.90E-04 5.80E-04 5.90E-04 5.10E-04 5.68E-04 4.00E-05 1.30E-02 2.80E-04 3.00E-04 2.90E-04 4.10E-04 3.20E-04 6.50E-05 1.59E-02 3.40E-04 2.80E-04 3.50E-04 3.50E-04 3.30E-04 3.50E-05 1.92E-02 1.80E-04 1.80E-04 1.70E-04 2.50E-04 1.95E-04 4.00E-05 2.22E-02 1.90E-04 1.80E-04 2.30E-04 2.00E-04 2.00E-04 2.50E-05 2.56E-02 1.20E-04 1.20E-04 1.20E-04 1.60E-04 1.30E-04 2.00E-05 2.89E-02 1.10E-04 1.20E-04 1.00E-04 1.50E-04 1.20E-04 2.50E-05 3.17E-02 8.00E-05 9.00E-05 8.00E-05 1.10E-04 9.00E-05 1.50E-05 3.54E-02 7.00E-05 8.00E-05 7.00E-05 1.00E-04 8.00E-05 1.50E-05 The distance in mm can be converted to m and the mass converted from g to kg. Table 3: Converted Data PROCESSED DATA Formula Sample Calculation F = mg F = 5.90 × 10 4 × 9.81 F = 5.79 × 10 3 N B = F I l ⊥ B = 5.79 × 10 3 0.600 × 0.0088 B = 1.054 T B axis = μ 0 4 π 2 μ d 3 B axis = 4 π × 10 − 7 4 π × 2 × 8.09 ( 0.00982 ) 3 B axis = 1.709 T ± = max − min 2 ± = 0.59 − 0.51 2 ± = 0.04 % ± = ± value × 100 % ± = 0.0005 0.600 × 100 % ± = 0.08 Three Line ± = | max gradient − min gradient 2 | ± = | − 1.81514 + 1.24403 2 | ± = 0.2856 %error = | measured − actual actual | × 100 %error = | 1.054 − 1.709 1.709 | × 100 ¿ 38.3% % ± B = % ±F + % ± I + % ±l ⊥ % ± B = 0.08% + 0.57% + 7.05% ¿ 7.70% Table 4: Sample Calculations IA2-SE 6

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The kg values are multiplied by 9.81 to obtain force in Newtons. Force F (N) ± 0.00005 Separation r (m) ± 0.0005 T1 T2 T3 T4 Average F (N) ± %± 9.82E-03 5.79E-03 5.69E-03 5.79E-03 5.00E-03 5.57E-03 3.92E-04 7.05 1.30E-02 2.75E-03 2.94E-03 2.84E-03 4.02E-03 3.14E-03 6.38E-04 20.3 1.59E-02 3.34E-03 2.75E-03 3.43E-03 3.43E-03 3.24E-03 3.43E-04 10.6 1.92E-02 1.77E-03 1.77E-03 1.67E-03 2.45E-03 1.91E-03 3.92E-04 20.5 2.22E-02 1.86E-03 1.77E-03 2.26E-03 1.96E-03 1.96E-03 2.45E-04 12.5 2.56E-02 1.18E-03 1.18E-03 1.18E-03 1.57E-03 1.28E-03 1.96E-04 15.4 2.89E-02 1.08E-03 1.18E-03 9.81E-04 1.47E-03 1.18E-03 2.45E-04 20.8 3.17E-02 7.85E-04 8.83E-04 7.85E-04 1.08E-03 8.83E-04 1.47E-04 16.7 3.54E-02 6.87E-04 7.85E-04 6.87E-04 9.81E-04 7.85E-04 1.47E-04 18.8 Table 5: Force data By substituting the appropriate values into B = F I l ⊥ , a set of field strength points can be calculated for each separation. The percentage and absolute uncertainties are also calculated. Separation r (m) Field Strength B (T) %± ± 0.00982 1.054 7.70 0.08 0.01303 0.595 21.0 0.12 0.01592 0.613 11.3 0.07 0.01921 0.362 21.2 0.08 0.02222 0.372 13.2 0.05 0.02557 0.242 16.1 0.04 0.02889 0.223 21.5 0.05 0.03174 0.167 17.3 0.03 0.03535 0.149 19.4 0.03 Table 6: Field Strength data A plot of B vs r can now be constructed: IA2-SE 7

0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.000 0.200 0.400 0.600 0.800 1.000 1.200 f(x) = 0 x^-1.5 R² = 0.97 Seperation r (m) Magnetic Field Strength B (T) Figure 3: B vs r Figure 3 indicates that B decreases as r increases, following an inverse relationship with a power of 1.5. T4 is highlighted red suggesting that it could be an outlier. Therefore, a new set of calculated B values with their appropriate uncertainties can be produced. Separation r (m) Field Strength B (T) %± ± 0.00982 1.090 1.50 0.016 0.01303 0.539 4.10 0.024 0.01592 0.601 11.5 0.070 0.01921 0.328 3.48 0.013 0.02222 0.372 13.2 0.049 0.02557 0.223 0.00 0.000 0.02889 0.204 9.74 0.022 0.03174 0.155 6.65 0.011 0.03535 0.136 7.47 0.011 Table 7: B calculations excluding T4 A plot of B vs r can be now be constructed, not including T4: IA2-SE 8

0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0 0.2 0.4 0.6 0.8 1 1.2 f(x) = 0 x^-1.56 R² = 0.95 Separation r (m) Magnetic Field Strength B (T) Figure 4: B vs r excluding T4 A relationship of B ∝ 1 r 1.56 is evident, however the vertical error bars have significantly decreased. Figure 4 can be linearised by applying a log-log analysis. The %± in log B will be assumed to equal to the %± in B from table 7. Log r (m) Log B (T) %± ± -2.008 0.037 1.50 -0.03 -1.885 -0.269 4.10 -0.08 -1.798 -0.221 11.48 -0.21 -1.716 -0.484 3.48 -0.06 -1.653 -0.430 13.15 -0.22 -1.592 -0.652 0.00 0.00 -1.539 -0.690 9.74 -0.15 -1.498 -0.810 6.65 -0.10 -1.452 -0.866 7.47 -0.11 Table 8: Linearised data A plot of log B vs log r can now be constructed: IA2-SE 9

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-2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 f(x) = 0 x − 0.07 R² = 0.08 Data Linear (Data) MaxSlope Linear (MaxSlope) MinSlope Linear (MinSlope) MaxWindow Log r (m) Log B (T) Figure 5: Log B vs Log r The linearisation clearly validates the relationship identified above, that B ∝ 1 r 1.56 . LINEST and three-line method data analysing is applied to figure 5. Linear analysis Gradient -1.56 Intercept -3.11 ± Gradient 0.13 ± Intercept 0.22 Table 9: LINEST and three-line method analysis IA2-SE 10 LINEST: log B = ( − 1.56 ± 0.13 m ) log r +(− 3.11 ± 0.22 m ) Three-line method: log B = ( − 1.56 ± 0.25 m ) log r +(− 3.11 ± 0.47 m ) Three-line method Max Trendline -1.87x - 3.69 Min Trendline -1.37x – 2.75 ± Gradient 0.25 ± Intercept 0.47

ANALYSIS OF DATA As seen in figure 4, B decreases from 1.1 T to 0.14 T as the distance of separation is increased from 0.01 to 0.035 following the law B ∝ 1 r 1.56 . The curve is reducing and shows a clear decrease in error bar size compared to figure 3. Compared to figure 2, the experimental curve aligns with the shape of the theoretical and furthermore the inverse power relationship identified lies within the expected range of B ∝ 1 r →B ∝ 1 r 3 . Figure 5 demonstrates a negative linear correlation between the data and validates a power of 1.56 - consistent with the relationship in figure 4. Furthermore, the size of the error bars generally increase as the distance is increased, especially for log r’s of -1.798, -1.653 and -1.539 (table 8). This trend could be a result of assuming the %± in log B is equal to the %± in B. LIMITATIONS OF EVIDENCE As per table 6, percentage uncertainties of 21.0% 21.2%, 21.5% appear particularly high, contributing to the size of the error bars in figure 3. Meanwhile, Trial 4 was excluded because it creates a level of uncertainty in the data. Figure 4 indicates that the points from 0.01 to 0.025 m align with the trendline, apparently reflecting random error in the data. Furthermore, there is a large difference in B between the first and second points, leaving fewer points where the curve is rapidly changing. The assumptions that the wire position would remain constant when separating the magnets and that the grade of the magnets was N-42 could have contributed to error within the data. The three-line method calculated a gradient ± of 0.25, whilst the linear analysis calculated a ± of 0.13 (Table 7). The linear analysis best represents the data since the points are closer to the trendline, whilst the three-line method uses the first and last points of the vertical error bars resulting in a large difference between the opposing gradients. IA2-SE 11

INTERPRETATION OF DATA Figure 4 indicated an inverse-power relationship between B and r that: B ∝ 1 r 1.56 . Furthermore, figure 5 confirmed that B generally decreases with an increase in separation of power 1.56 as validated by equation: log B = ( − 1.56 ± 0.13 m ) log r +(− 3.11 ± 0.22 m ) The inverse power relationship obtained in this experiment lies within the expected range of B ∝ 1 r →B ∝ 1 r 3 and the shape of the curve in figure 4 aligns with that of the theoretical (figure 2). The exclusion of trial four ensured less uncertainty within the data. However, some points did not fit the curve and, most significantly, fewer points were at the start of the curve, where it is rapidly changing. However, these limitations would not have a large effect on the relationship of B ∝ 1 r 1.56 . EVALUATION OF METHOD Figure 4 depicted a large difference between 0.01 and 0.015 m resulting in less data at the top of the curve. This limitation effects the ability to fit the trendline effectively as more points are needed for where it is rapidly changing. This may be due to not using thinner plates, since each point is equally separated horizontally (figure 4), preventing testing of a broader range of separations. Random error was observed regarding the points that did not fit the curve properly (figure 4). This may have been caused by not keeping the wire at the centre of the field when re-adjusting the separation, which may suggest the data to be unreliable. Considering the overall outcome, the data could likely be valid as the relationship identified falls within the expected range of B ∝ 1 r →B ∝ 1 r 3 . Furthermore, the shape of the experimental curve aligns with that of the theoretical and the linearisation of figure 4 validated a power of 1.56. IA2-SE 12

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CONCLUSION In conclusion, B and r satisfy a power relationship where B ∝ 1 r 1.56 . Furthermore, figure 5 validated the power determined in figure 4 and identified that B generally decreases as separation is increased by equation: log B = ( − 1.56 ± 0.13 m ) log r +(− 3.11 ± 0.22 m ) The determined inverse power relationship fell between the expected range of B ∝ 1 r →B ∝ 1 r 3 and the shape of the experiment curve aligned with that of the theoretical. Furthermore, trial four was excluded due to the decrease in uncertainty seen between figure 3 and 4. Limitations regarding the lack of points at the start of the curve and variation of the points from the curve may have caused error in the data, however the influence of these limitations would not effect the outcome of this experiment. Hence, as the results align with both theory and expectations, the research question has been effectively responded to. IMPROVEMENTS AND EXTENSIONS This experiment could be improved by using thinner plates to separate the magnets, which allows a greater range of separations to be tested and ensures that all points are equally spread across the curve. The experiment could also introduce a scale on the separator to ensure the position of the wire is kept constant in order to reduce random error. An extension could be using thin cylindrical disk magnets to investigate the relationship between B and r with only two poles. Since the type of magnets was kept constant in this experiment, changing the size would provide new insight into the interaction of fields. BIBLIOGRAPHY K&j Magnets. (2021). K&J Magnetics - Magnet Calculator. Retrieved March 28, 2021 from https://www.kjmagnetics.com/calculator.asp IA2-SE 13

Texas Gateway. (2021). 20.1 Magnetic Fields, Field Lines, and Force | Texas Gateway. Retrieved March 28, 2021 from https://www.texasgateway.org/resource/201- magnetic-fields-field-lines-and-force Vernier. (2020). The magnetic field of a permanent magnet. Vernier. https://www.vernier.com/experiment/pwv-31_the-magnetic-field-of-a-permanent-magnet/. Alini, A., Champion, N., & Kuhn t. (2018). Nelson QScience (2nd ed). Ford Rachel. IA2-SE 14